Wrap cylinder around sphere
I think the quickest way to deal with this issue would be to load the reference image behind the mesh, tab into Edit mode with Proportional editing on and quickly move the tube into place in sections. With proportional editing on you won't have to move every vertices. If you find the proportional editing is distorting your tube shape just decrease the strength/area of the tool's effect.

Also, I don't think you're going to be able to have a perfectly curved innner edge on the curve and have it snapped exactly to the sphere's surface because of the hard angles of the sphere (see screen shot). This is because you still want the curve to be smooth which will require some subsurf to be applied.